Exact and asymptotic properties of $\delta$-records in the linear drift model

TL;DR

This paper investigates the properties and asymptotic behavior of $oldsymbol{ ext{ extdelta}}$-records in the Linear Drift Model, providing analytical results, solving a conjecture, and applying findings to climate data.

Contribution

It introduces the study of $ ext{ extdelta}$-records in the LDM, analyzing their probability, correlation, and asymptotic distribution, and applies results to real climate data.

Findings

Derived analytical properties of $ extdelta$-record probabilities.

Established conditions for Gaussian convergence of record counts.

Solved a conjecture on total records in LDM with negative drift.

Abstract

The study of records in the Linear Drift Model (LDM) has attracted much attention recently due to applications in several fields. In the present paper we study $\delta$-records in the LDM, defined as observations which are greater than all previous observations, plus a fixed real quantity $\delta$. We give analytical properties of the probability of $\delta$-records and study the correlation between $\delta$-record events. We also analyse the asymptotic behaviour of the number of $\delta$-records among the first $n$ observations and give conditions for convergence to the Gaussian distribution. As a consequence of our results, we solve a conjecture posed in J. Stat. Mech. 2010, P10013, regarding the total number of records in a LDM with negative drift. Examples of application to particular distributions, such as Gumbel or Pareto are also provided. We illustrate our results with a real data set of summer temperatures in Spain, where the LDM is consistent with the global-warming phenomenon.